# Closed convex hull of set of measurable functions, Riemann-measurable functions and measurability of translations

Annales de l'institut Fourier (1982)

- Volume: 32, Issue: 1, page 39-69
- ISSN: 0373-0956

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topTalagrand, Michel. "Closed convex hull of set of measurable functions, Riemann-measurable functions and measurability of translations." Annales de l'institut Fourier 32.1 (1982): 39-69. <http://eudml.org/doc/74530>.

@article{Talagrand1982,

abstract = {Let $G$ be a locally compact group. Let $L_t$ be the left translation in $L^\infty (G)$, given by $L_tf(x) = f(tx)$. We characterize (undre a mild set-theoretical hypothesis) the functions $f\in L^\infty (G)$ such that the map $t\rightarrow L_tf$ from $G$ into $L^\infty (G)$ is scalarly measurable (i.e. for $\varphi \in L^\infty (G)^*$, $t\rightarrow \varphi (L_tf)$ is measurable). We show that it is the case when $t\rightarrow \theta (L_ft)$ is measurable for each character $\theta $, and if $G$ is compact, if and only if $f$ is Riemann-measurable. We show that $t \rightarrow L_tf$ is Borel measurable if and only if $f$ is left uniformly continuous.Some of the measure-theoretic tools used there have independent interest. For example, if a set of measurable functions on $[0,1]$ is separable and point-wise relatively compact, the same is true of its convex hull.},

author = {Talagrand, Michel},

journal = {Annales de l'institut Fourier},

keywords = {locally compact group; left translation; scalarly measurable; Borel measurable; left uniformly continuous},

language = {eng},

number = {1},

pages = {39-69},

publisher = {Association des Annales de l'Institut Fourier},

title = {Closed convex hull of set of measurable functions, Riemann-measurable functions and measurability of translations},

url = {http://eudml.org/doc/74530},

volume = {32},

year = {1982},

}

TY - JOUR

AU - Talagrand, Michel

TI - Closed convex hull of set of measurable functions, Riemann-measurable functions and measurability of translations

JO - Annales de l'institut Fourier

PY - 1982

PB - Association des Annales de l'Institut Fourier

VL - 32

IS - 1

SP - 39

EP - 69

AB - Let $G$ be a locally compact group. Let $L_t$ be the left translation in $L^\infty (G)$, given by $L_tf(x) = f(tx)$. We characterize (undre a mild set-theoretical hypothesis) the functions $f\in L^\infty (G)$ such that the map $t\rightarrow L_tf$ from $G$ into $L^\infty (G)$ is scalarly measurable (i.e. for $\varphi \in L^\infty (G)^*$, $t\rightarrow \varphi (L_tf)$ is measurable). We show that it is the case when $t\rightarrow \theta (L_ft)$ is measurable for each character $\theta $, and if $G$ is compact, if and only if $f$ is Riemann-measurable. We show that $t \rightarrow L_tf$ is Borel measurable if and only if $f$ is left uniformly continuous.Some of the measure-theoretic tools used there have independent interest. For example, if a set of measurable functions on $[0,1]$ is separable and point-wise relatively compact, the same is true of its convex hull.

LA - eng

KW - locally compact group; left translation; scalarly measurable; Borel measurable; left uniformly continuous

UR - http://eudml.org/doc/74530

ER -

## References

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- [2] D.H. FREMLIN, Pointwise compact sets of measurable functions, Manuscripta Math., 15 (1975), 219-242. Zbl0303.28006MR51 #8801
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- [5] A. and C. IONESCU-TULCEA, Topics in the Theory of Liftings, Springer Verlag, 1979.
- [6] A. and C. IONESCU-TULCEA, On the existence of a lifting commuting with the left translations of an arbitrary locally compact group, Proceedings Fifth Berkeley Symposium on Math. and Probability, 63-97. Zbl0201.49202
- [7] J.M. ROSENBLATT, Invariant means for the bounded measurable functions on a non-discrete locally compact group, Math. Annalen, 220 (1976), 219-228. Zbl0305.43002MR53 #1164
- [8] W. RUDIN, Homomorphisms and translations in L∞(G), Advances in Math., 16 (1975), 72-90. Zbl0297.22009MR51 #3797
- [9] J.R. SCHOENFIELD, Martin's axiom, Amer. Math Monthly, 82 (1975), 610-617. Zbl0314.02069MR51 #10087
- [10] B. WELLS, Homomorphisms and translates of bounded functions, Duke Math. J., 41 (1974), 35-39. Zbl0281.28004MR49 #1014

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